I have some problems interpreting a categorical multilevel SEM model. It is a random intercept model with the same measurement part on both levels.
I believe, on the within-level factor loadings would represent the within-level components of a latent variable (corrected for the group-specific means). If the observed indicators were continuous, I would assume that these are the between-level components of the latent variable (random intercepts).
Now, how do I interpret this if my observed variables are categorical? Are the factor loadings still random intercepts or it is now thresholds?
Can I assume that if the coefficient for a given threshold is not significant, there is no variation in the proportion of responses (intercepts?) to respective categories across groups? This would make sense, since the measurement equivalence testing of the same construct across groups points at the same results - some thresholds are invariant, where's the other need to be set free so the partial threshold model would fit the data.
Could you also point me at some literature on categorical multilevel SEM and particularly the interpretation part? I am not sure whether there are now adopted standards for a scientific publication (say, APA guidelines) for this type of models. Particularly, would it be appropriate to report the residual variances?
The loadings are not random in Mplus multilevel factor analysis, that is, they don't vary across cluster units. The intercepts are random. This is the same for continuous and categorical outcomes. The random intercepts can be seen as capturing the cluster variation in each factor indicator and this variation can be expressed by a between-level factor. With equality of factor loadings across within and between, one can decompose the factor variances in within and between parts. Unfortunately there isn't much written yet about multilevel factor analysis, especially for categorical outcomes. You can watch what we say about it in the Topic 7 video on our web site.
I want to clarify a few points. On the between level I get the intercepts for the mediating variables, since I have a path structure there. I believe I then can assume that they capture the random variation in the intercepts on the group level (time, in my case).
However, since I have categorical observed indicators, I do not get the intercept coefficients for those variables in the output (although I believe this is indeed a part of the output in the multi-group SEM routine with categorical indicators). Instead, I only get the thresholds for each observed categorical indicator. Does the interpretation you offer still hold for the thresholds? Otherwise, I am still not sure how to interpret them in this case.
On top of that, could I interpret the R2 in the multilevel SEM model w/ categorical indicators similar to the R2 in the OLS regression, or should I treat it as the ball-park estimate like pseudo-R2?
Again, I appreciate your time and feedback, Dmitriy
I think you are saying that you don't get residual variances on between for each categorical indicator. Depending on the model, you may have to mention these residual variances on between for them not to be fixed at zero as the default.
R2 for categorical indicators is the usual R2 on between. On within it is using the fixed residual variance (at 1 for probit and pi-sq/3 for logit) that has been advocated by McKelvey & Savoina article and in the Snijders and Bosker book.